Monotone decompositions of continua not separated by any subcontinua

Eldon J. Vought
1974 Transactions of the American Mathematical Society  
Let M be a compact, metric continuum that is separated by no subcontinuum. If such a continuum has a monotone, upper semicontinuous decomposition, the elements of which have void interior and for which the quotient space is a simple closed curve, then it is said to be of type A'. It is proved that a bounded plane continuum is of type A' if and only if M contains no indecomposable subcontinuum with nonvoid interior. In E} this condition is not sufficient and an example is given to illustrate
more » ... n to illustrate this. However, it is shown that if M is hereditarily decomposable then M is of type A'. Next, a condition is given that characterizes continua of type A'. Also the structure of the elements in the decomposition of a continuum of type A' is discussed and the decomposition is shown to be unique. Finally, some consequences of these results and some remarks are given. Presented to the Society, November 18,1972 under the title Monotone decompositions of continua into simple closed curves and generalized simple closed curves;
doi:10.1090/s0002-9947-1974-0341438-x fatcat:qquttlp4ojhmriyouzzovtqcl4